Method for determining dynamic transmission error of gear

ABSTRACT

A gear dynamic transmission error determination method includes calculating a collision time interval via parameter values such as the angular velocity of an input shaft measured by an angle coder, a drag torque of an output shaft provided by a magnetic powder brake, a gear backlash, and angular displacements of the input shaft and the output shaft; and comparing a relative displacement contained with the gear backlash, until a value of the relative displacement falls within an appropriate interval and meets requirements, the result obtained being namely a transmission error with relatively high accuracy.

TECHNICAL FIELD

The present invention relates to the field of machine manufacturing and controlling, and more particularly, to a method for determining dynamic transmission error of gear considering a gear backlash.

BACKGROUND

In all mechanical drive, gear transmission has characteristics of steady drive, accuracy of drive ratio, high efficiency, long service life or the like, so that the gear transmission becomes the most important and widely used mechanism transmission High speed steady and precise gear transmission is a national key scientific and technological project. A transmission error has been widely accepted and considered as an excitation source for vibration and noise of a gear system because the transmission error affects force and velocity changes in the gear transmission. At the same time, the gear dynamic transmission error is also a very important part in gear testing, so that accurate testing and calculation of the transmission error are of great significance to gear manufacturing and the researches on gear dynamics. The State Intellectual Property Office of the People's Republic of China disclosed a patent documentation on 1 Oct., 2008, with a Publication Patent No. of CN101275881, and titled Small Mode Number Gear transmission Error Measuring Method, including the steps of firstly connecting a measured gear, a driver, a first goniometric coder, a first shaft joint and a first electric motor in successive, then connecting a measuring gear, a flexible shaft joint, a second goniometric coder, a second shaft joint and a second electric motor in successive, the measured gear and the measuring being meshed in a single side and being driven by the two electric motors respectively, detecting an angular displacement Φ2 of the measuring gear relative to the second goniometric coder, an angular displacement Φ1 of a measuring gear shaft system and an angular displacement Φ′ of a measured gear shaft system, and obtaining a transmission error of a small mode number gear after calculation. However, due to gear machining errors, installation errors, lubrication, modification and other factors, an offset of a tooth profile surface relative to an ideal tooth profile position will lead to a gear backlash (FIG. 1), so that positions of a meshing-in point and a meshing-out point will deviate from a theoretical meshing point when the gear is meshed in or meshed out, to produce corner meshing and cause a collision impact between meshing gear surfaces. Such collision impact plays a key role in smoothness and accuracy of high-speed gear transmission. For accuracy drive, trace idle stroke and impact occur during positive and reverse rotating, which affect the drive accuracy. Obviously, the gear backlash directly affects a gear system to obtain the transmission error. However, the influence of the gear backlash on the transmission error is not considered in this solution, so that a larger deviation will occur between a final calculation result and an actual value.

At the same time, the numerical calculation of the transmission error of the gear system becomes an increasingly prominent problem. At present, transmission error calculation methods stay in the traditional methods, such as: a Newmark integral method, a Runge-Kutta integral method, a Gear method and an improved Gear method (Wstiff method and Dstiff method). The above calculation methods have different emphasis on the transmission error calculation methods. Some methods focus more on grasping the calculation accuracy; while some methods have a faster calculation velocity. The traditional calculation methods usually fail to control the distribution of calculation time and calculation accuracy perfectly, often bring an enormous calculation quantity in case that the accuracy requirement is higher, resulting in a long calculation time; and will decrease the calculation accuracy in case of guaranteeing the calculation time, so that the transmission error of the gear system cannot be described and forecasted correctly.

SUMMARY

The present invention mainly solves a technical problem of the prior art that a transmission error is measured inaccurately due to a gear backlash during actual measurement, and a problem of being difficult to balance a calculation accuracy and a calculation time by calculating a gear dynamic transmission error through theoretical simulation. Under the premise of taking the influence of the gear backlash on the transmission error into consideration, a calculation method that combines the actual measurement with the theoretical simulation and can quickly calculate the gear dynamic transmission error in case of guaranteeing certain calculation accuracy is proposed.

The above technical problems are mainly solved by the present invention through the following technical solutions: a method for determining dynamic transmission error of gear includes the following steps of:

A. obtaining parameter values, the parameter values including an angular velocity of an input shaft measured by an angle coder and a drag torque of an output shaft provided by a magnetic powder brake;

B. calculating a collision time interval Δt_(i) between two gear teeth;

C. comparing the collision time interval Δt_(i) with a time-step Δt, entering step D if Δt is less than Δt_(i); and entering step G if Δt is greater than or equal to Δt_(i);

D. calculating a relative displacement x(n+1) by taking the time-step Δt as a step, if an absolute value |x(n+1)| of the relative displacement x(n+1) is less than or equal to a gear backlash L, entering step J; and if the absolute value |x(n+1)| of the relative displacement x(n+1) is greater than the gear backlash L, shortening the time-step Δt and then entering step E. comparing the time-step Δt with a minimum time-step t_(min), entering step D if the time-step Δt is greater than or equal to the minimum time length t_(min); and entering step F if the time-step Δt is less than the minimum time-step t_(min);

F. calculating a relative displacement t_(min) by taking the minimum time-step x(n+1)=Lsign({dot over (x)}(t(n))) as a step, {dot over (x)}(n+1)=0, and then entering step J;

G. calculating the relative displacement x(n+1) by taking the collision time interval Δt_(i) as the step, if an absolute value |x(n+1)| of the relative displacement x(n+1) is less than or equal to L−ε, entering step J; if the absolute value |x(n+1)| of the relative displacement x(n+1) is less than or equal to L+ε and the absolute value |x(n+1)| of the relative displacement x(n+1) is greater than L−ε, entering step I; and if the absolute value |x(n+1)| of the relative displacement is greater than L+ε, decreasing the collision time interval Δt_(i), and then entering step H;

H. comparing the collision time interval Δt_(i) with the minimum time-step t_(min), if the collision time interval Δt_(i) is greater than or equal to the minimum time-step, returning to step G; if the collision time interval Δt_(i) is less than the minimum time-step t_(min), taking the minimum time-step as the step, i.e., t(n+1)=t(n)+t_(min) and then entering step I;

I. calculating the relative displacement x(n+1)=L sign({dot over (x)}(t(n))), increasing the number of the step n by 1, then taking a collision time constant ∈_(Δt) as a step, i.e., t(n+1)=t(n)+∈_(Δt), calculating a relative velocity as: {dot over (x)}(n+1)=−e{dot over (x)}(n), calculating the relative displacement as: x(n+1)=x(n), and then entering step J; and

J. judging whether t(n+1) is greater than M×T_(f), if yes, finishing calculation, and if not, increasing the number of step n by 1 and then returning to step B;

ε being a minimum error, M is a number of the period, T_(f) being a period of the excitation, t(n) being a t(n)^((th)) time sampling point, and x(n) being an n^((th)) relative displacement, if an original point being located above a sign, it representing a derivation for time, one original point representing a first-order derivative, two original points representing second-order derivative, and sign being a sign function. The relative displacement x(n+1), relative velocity {dot over (x)}(n+1) and time t(n+1) of the M periods finally obtained can reflect the gear dynamic transmission error preferably. Through the transmission error obtained, the accuracy of a gearbox may be evaluated.

When measuring the actual gear set, various parameter values are measured in a conventional manner or set by a user; during analogue simulation calculation, the parameter values are manually set or automatically set by a computer. The parameter values include L, ε, T_(f), R_(bg), θ_(g), θ_(p), the mean value of the external load moment, the rotational inertia of the driven wheel, the fluctuation portion of the external load moment, ω, t₀, the meshing stiffness at the front and lateral sides of the gear pair caused by the oil film, and the meshing damping at the front and lateral sides of the gear pair caused by the oil film, or the like.

Preferably, a calculation mode of the collision time interval in step B is as follows:

${{\Delta \; t_{i}} = \frac{{- b} \pm \sqrt{b^{2} - {4\; {ac}}}}{2\; a}},$

wherein, a=(R_(bg)T_(m)+R_(bg)T_(p) cos(ωt₀)−R_(bg)kx(t₀)−R_(bg)c{dot over (x)}(t₀))/2,b={dot over (x)}(t₀), c=x(t₀)−Lsign({dot over (x)}(t₀)), Δt_(i) is a minimum positive real value of two values, R_(bg) is a base radius of a driven wheel, R_(bp) is a base radius of a driving wheel, T_(m) is a ratio of a mean value of an external load moment to a rotational inertia of the driven wheel, T_(p) is a ratio of a fluctuation portion of the external load moment to the rotational inertia of the driven wheel, ω is a frequency of the external load moment, t₀ is an initial system time, k is a ratio of a sum of meshing stiffness at front and lateral contact sides of a gear pair caused by an oil film to the rotational inertia of the driven wheel, and c is a ratio of a sum of meshing damping at the front and lateral contact sides of the gear pair caused by the oil film to the rotational inertia of the driven wheel.

Preferably, Δt_(i)<Δt is directly judged if Δt_(i) has no positive real value while calculating the collision time interval Δt_(i).

Preferably, a specific operation of shortening the time-step in step D is decreasing Δt² from Δt.

Preferably, a specific operation of shortening the collision time interval in step D is decreasing Δt_(i) ² from Δt_(i).

Substantial effects brought by the present invention are that the influence of the gear backlash on the transmission error is considered, and the calculation time is shortened as much as possible in the case of guaranteeing certain calculation accuracy, so that more preferable distribution of the calculation accuracy and calculation time is realized.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an enlarged drawing of a gear meshing area of the present invention;

FIG. 2 is a schematic diagram of a parameter measurement system of the present invention;

FIG. 3 is a schematic diagram of assembling a driving wheel and a driven wheel according to a standard center distance of the present invention;

FIG. 4 is a schematic diagram of a judgment situation when the driving wheel is adhered to and linked with the driven wheel of the present invention;

FIG. 5 is a schematic diagram of a situation when Δt≧Δt during collision between teeth of the present invention;

FIG. 6 is a schematic diagram of a situation of using ε to make the collision between gear teeth precise again according to the present invention; and

FIG. 7 is a flow chart of a determination method of the present invention; in figures: 1 refers to electric motor, 2 refers to pinion gear; 3 refers to input shaft; 4 refers to angle coder; 5 refers to large gear; 6 refers to output shaft; and 7 refers to magnetic powder brake.

DETAILED DESCRIPTION

The technical solutions of the present invention will be further described hereinafter with reference to the embodiments and the drawings.

Embodiment: FIG. 2 is a system for measuring an actual gear set, which includes a motor 1, a driving gear 2, an input shaft 3, an angular coder 4, a driven wheel 5, an output shaft 6 and a magnetic powder brake 7. The driving wheel 2 and the driven wheel 5 are assembled according to a standard center distance, as shown in FIG. 3.

FIG. 1 is an enlarged drawing of a gear meshing area.

A transmission error (relative displacement) of a gear is a difference value between a position of a tooth profile on the driven wheel measured along a direction of a meshing line and related to drive characteristic during actual meshing and a position where the tooth profile should be located under an ideal condition, which may be defined as follows:

x(t)=R _(bp)θ_(p)(t)−R _(bg)θ_(g)(t)   (1)

wherein, R_(bg) is a base radius of the driven wheel, R_(bp) is a base radius of the driving wheel, θ_(g) is an angular displacement of the driven wheel, θ_(p) is an angular displacement of the driving wheel, and the angular displacement may also be measured by the angular coder.

Angular coder: measuring the angular velocity {dot over (θ)}_(p) of the input shaft in FIG. 2; Magnetic powder brake: providing a drag torque for the output shaft, wherein the drag torque is:

T _(g) =T _(m) +T _(p) cos(ωt)   (2)

wherein, T _(m) and T _(p) are the mean value of the external load moment and the moment of the fluctuation portion respectively; ω is a frequency of the drag torque. When |x(t)|<L, a dynamic equation of the gear system is established according to Alembert principle:

I _(g){umlaut over (θ)}_(g)(t)−(c ₁ +c ₂){dot over (x)}(t)−(k ₁ +k ₂)x(t)=− T _(g)   (3)

wherein, I_(g) is a rotational inertia of a large gear; k₁ and c₁ are meshing stiffness and meshing damping at the front contact side of the gear pair caused by the oil film respectively; and T _(m) and T _(p) are the mean value of the external load moment and the moment of the fluctuation portion respectively. Both sides of the Formula (3) are divided by the rotational inertia I_(g) at the same time to obtain:

{umlaut over (θ)}_(g)(t)−c{dot over (x)}(t)−kx(t)=−T _(m) −T _(p) cos(ωt)   (4)

Here, c=(c₁+c₂)/I_(g), k=(k₁+k₂)/I_(g), T_(m)=T _(m)/I_(g)and T_(p)=T _(p)/I_(g). According to the formula (4), it can be obtained:

{umlaut over (θ)}_(g)(t)=c{dot over (x)}(t)+kx(t)−T _(m) −T _(p) cos(ωt)   (5)

From time t₀ to time t, the angular velocity and the angular displacement of the big gear wheel may be obtained by performing integral on the formula (5).

$\begin{matrix} {{{\overset{.}{\theta}}_{g}(t)} = {{{\overset{.}{\theta}}_{g}\left( t_{0} \right)} - {T_{m}\left( {t - t_{0}} \right)} - {T_{p}\frac{{\sin \left( {\omega \; t} \right)} - {\sin \left( {\omega \; t_{0}} \right)}}{\omega}} + {c\left( {{x(t)} - {x\left( t_{0} \right)}} \right)} + {k\left( {{{x\left( t_{0} \right)}\left( {t - t_{0}} \right)} + {{\overset{.}{x}\left( t_{0} \right)}\frac{\left( {t - t_{0}} \right)^{2}}{2}}} \right)}}} & (6) \\ {{\theta_{g}(t)} = {{\theta_{g}\left( t_{0} \right)} + {{{\overset{.}{\theta}}_{g}\left( t_{0} \right)}\left( {t - t_{0}} \right)} - {T_{m}\frac{\left( {t - t_{0}} \right)^{2}}{2}} + {T_{p}\frac{{\cos \left( {\omega \; t} \right)} - {\cos \left( {\omega \; t_{0}} \right)}}{\omega^{2}}} + {T_{p}\frac{\sin \left( {\omega \; t_{0}} \right)}{\omega}\left( {t - t_{0}} \right)} + {{{kx}\left( t_{0} \right)}\frac{\left( {t - t_{0}} \right)^{2}}{2}} + {K\; {\overset{.}{x}\left( t_{0} \right)}\frac{\left( {t - t_{0}} \right)^{3}}{6}} + {c\; {\overset{.}{x}\left( t_{0} \right)}\frac{\left( {t - t_{0}} \right)^{2}}{2}}}} & (7) \\ {{\theta_{p}(t)} = {{\theta_{p}\left( t_{0} \right)} + {{{\overset{.}{\theta}}_{p}\left( t_{0} \right)}\left( {t - t_{0}} \right)}}} & (8) \end{matrix}$

The formula (7) and the formula (8) are substituted into the formula (1) to obtain:

$\begin{matrix} {{x(t)} = {{x\left( t_{0} \right)} + {{\overset{.}{x}\left( t_{0} \right)}\left( {t - t_{0}} \right)} - {R_{bg}\begin{pmatrix} {{T_{p}\frac{{\cos \left( {\omega \; t} \right)} - {\cos \left( {\omega \; t_{0}} \right)}}{\omega^{2}}} + {T_{p}\frac{\cos \left( {\omega \; t_{0}} \right)}{\omega}\left( {t - t_{0}} \right)} +} \\ {{\left( {{{kx}\left( t_{0} \right)} + {c\; {\overset{.}{x}\left( t_{0} \right)}} - T_{m}} \right)\frac{\left( {t - t_{0}} \right)^{2}}{2}} + {k\; {\overset{.}{x}\left( t_{0} \right)}\frac{\left( {t - t_{0}} \right)^{3}}{6}}} \end{pmatrix}}}} & (9) \\ {{\overset{.}{x}(t)} = {{\overset{.}{x}\left( t_{0} \right)} - {R_{bg}\begin{pmatrix} {{c\left( {{x(t)} - {x\left( t_{0} \right)}} \right)} + {k\left( {{{x\left( t_{0} \right)}\left( {t - t_{0}} \right)} + {{\overset{.}{x}\left( t_{0} \right)}\frac{\left( {t - t_{0}} \right)^{2}}{2}}} \right)} -} \\ {{T_{m}\left( {t - t_{0}} \right)} - {T_{p}\frac{{\sin \left( {\omega \; t} \right)} - {\sin \left( {\omega \; t_{0}} \right)}}{\omega}}} \end{pmatrix}}}} & (10) \end{matrix}$

When |x(t)|=L, an elastic collision occurs to the gear system, and the dynamic equation of the system at this moment is:

x(t+∈ _(Δt))=x(t)′{dot over (x)}(t+∈ _(Δt))=−e{dot over (x)}(t)   (11)

wherein, e is a collision coefficient, and a collision time constant ∈_(Δt) is an extremely small positive number. During collision, the transmission error may be represented as:

x(t ₀ +Δt _(t))=Lsign({dot over (x)}(t ₀))   (12)

Here, Δt_(i)=t_(i)−t₀ is a collision time interval; if sign({dot over (x)}(t₀))>0, it represents that the collision occurs to the front side of the gear tooth, at this moment, the driven wheel rotates under the driving of the driving wheel. If sign({dot over (x)}(t₀))<0, it represents that the collision occurs on the lateral side of the gear tooth, at this moment, the driving wheel rotates under the driving of the driven wheel. The formula (9) is combined with the formula (12) to obtain:

$\begin{matrix} {{{Lsign}\left( {\overset{.}{x}\left( t_{0} \right)} \right)} = {{x\left( t_{0} \right)} + {{\overset{.}{x}\left( t_{0} \right)}\Delta \; t_{i}} - {R_{bg}\begin{pmatrix} {{T_{p}\frac{{\cos \left( {\omega \; t} \right)} - {\cos \left( {\omega \; t_{0}} \right)}}{\omega^{2}}} + {T_{p}\frac{\sin \left( {\omega \; t_{0}} \right)}{\omega}\Delta \; t_{i}} +} \\ {{\left( {{{kx}\left( t_{0} \right)} + {c\; {\overset{.}{x}\left( t_{0} \right)}} - T_{m}} \right)\frac{\Delta \; t_{i}^{2}}{2}} + {k\; {\overset{.}{x}\left( t_{0} \right)}\frac{\Delta \; t_{i}^{3}}{6}}} \end{pmatrix}}}} & (13) \end{matrix}$

Second-order Taylor expansion is conducted near the small quantity of Δt_(i) on left and right sides of the formula (13) at the same time, and an infinitely small quantity having an order higher than Δt_(i) ² is ignored, to obtain:

$\begin{matrix} {{{Lsign}\left( {\overset{.}{x}\left( t_{0} \right)} \right)} \approx {{x\left( t_{0} \right)} + {{\overset{.}{x}\left( t_{0} \right)}\Delta \; t_{i}} - {\frac{R_{bg}}{2}\left( {{{kx}\left( t_{0} \right)} + {c\; {\overset{.}{x}\left( t_{0} \right)}} - T_{m} - {T_{p}{\cos \left( {\omega \; t_{0}} \right)}}} \right)\Delta \; t_{i}^{2}}}} & (14) \end{matrix}$

The following may be determined according to the above formula:

$\begin{matrix} {{\Delta \; t_{i}} = \frac{{- b} \pm \sqrt{b^{2} - {4\; {ac}}}}{2\; a}} & (15) \end{matrix}$

wherein, a=(R_(bg)T_(m)+R_(bg)T_(p) cos(ωt₀)−R_(bg)kx(t₀)−R_(bg)c{dot over (x)}(t₀))/2, b={dot over (x)}(t₀) and c=x(t₀)−Lsign({dot over (x)}(t₀)). Only a minimal positive real value of Δt_(i) is taken according to a physical significance. If Δt_(i) has no minimal positive real value, it is deemed that Δt<Δt_(i).

As shown in FIG. 7, Δt_(i) is compared with Δt to determine next time-step, if Δt<Δt_(i), then the relative displacement x(n+1) is calculated by the time-step Δt.

If |x(n+1)|<L, the time Δt_(i) of the next moment is calculated. If |x(n+1)|>L, it indicates that the relative displacement is greater than the gear backlash, and the time-step Δt shall be shortened according to formula Δt=Δt−Δt² (16).

If the shortened time-step is greater than the minimum time-step t_(min) set by the system according to the formula (16), then the relative displacement x(n+1) is recalculated. Otherwise, the minimum time-step t_(min) is taken as a step by the system to calculate the relative displacement x(n+1). At this moment, the large gear and the pinion gear of the system are adhered and linked (as shown in FIG. 4). At this moment, it may be defined according to the physical significance thereof as follows:

x(n+1)=Lsign({dot over (x)}(t(n)))′{dot over (x)}(n+1)=0   (17)

If Δt≧Δt_(i) (as shown in FIG. 5), then Δt_(i) is served as the next step to calculate next relative displacement and velocity. Similarly, x(n+1) needs to be judged, if |x(n+1)|>L+ε, it proves that the relative displacement x(n+1) herein goes beyond an accuracy range, as shown in FIG. 6, and Δt_(i) needs to be decreased here, so as to guarantee the accuracy within an allowable error range, and a formula of decreasing Δt_(i) is as follows:

Δt _(i) =Δt _(i) −Δt _(i) ²   (18)

Δt_(i) in the formula (18) is the step, to recalculate x(n+1) and {dot over (x)}(n+1). If |x(n+1)|>L+ε at this moment, the formula (18) is repeated until Δt_(i)<t_(min), then t_(min) is taken as a step by the system to calculate the relative displacement and the relative velocity. At this moment, a collision occurs to the system, and according to the actual physical significance of collision, set:

x(n+1)=Lsign({dot over (x)}(t(n)))′  (19)

In order to guarantee that the relative displacement values of the system before and after collision are equal, a small quantity of time is introduced herein, i.e., a collision time constant ∈_(Δt), so that ∈_(Δt) is taken as a step by the system, i.e.,:

t(n+1)=t(n)+∈_(Δt)   (20)

The relative displacement and the relative velocity are recalculated, and the relative displacement and relative velocity after the collision are as follows:

{dot over (x)}(n+1)=−e{dot over (x)}(n), x(n+1)=x(n)   (21)

If the formula |x(n+1)|>L+ε is not satisfied, but |x(n+1)|>L −ε is satisfied, a collision occurs to the system, and formulas (19), (20) and (21) are repeated at this moment.

If neither |x(n+1)|>L+ε nor |x(n+1)|>L−ε is satisfied, it illustrates that no collision occurs to the system, and the displacement and the velocity are recalculated at this moment.

Through continuous circulation in such a way, a large number of x(n+1), {dot over (x)}(n+1), t(n+1) and other data may be recorded. During the entire course of operation, if t(n+1)<M*T_(f), then the relative displacement and relative velocity obtained are substituted into the formula (15) to recalculate Δt_(i). If t(n+1)>M*T_(f), then the operation is stopped, thus being capable of obtaining the relative displacement (transmission error), the relative velocity (velocity of transmission error) and the time data of M periods. M is a number of the period of the excitation, T_(f) is a period of the excitation, and the excitation is provided by the magnetic powder brake.

In conclusion, according to the method for determining dynamic transmission error of gear of the present invention, the collision time interval is calculated via the parameter values such as the angular velocity of the input shaft 3 measured by the angle coder 4, the drag torque of the output shaft 6 provided by the magnetic powder brake 7, the gear backlash, and angular displacements of the input shaft 3 and the output shaft 6; and the relative displacement obtained is compared with the gear backlash, until the value of the relative displacement falls within an appropriate interval and meets requirements, the result obtained being namely the transmission error with relatively high accuracy. By considering the influence of the gear backlash on the transmission error, the accuracy of a transmission error determination result is guaranteed, meanwhile, the accuracy and time are also well balanced, and the method is applicable to the measurement, calculation and analogue simulation calculation of an actual gear set.

The specific embodiments described herein merely illustrate the spirit of the present invention. Those killed in the art may figure out various modifications or supplements or replacement in a similar mode on the specific embodiments described without departing from the spirit of the invention or going beyond the scope defined by the claims appended.

Although such terms as relative displacement and collision time interval are frequently used herein, this does not exclude the possibility of using other terms. These terms are merely used for describing and explaining the essence of the present invention more conveniently; and explaining the terms into any additional limitation departs from the spirit of the present invention. 

1. A method for determining dynamic transmission error of gear, comprising: A. obtaining parameter values, the parameter values comprising an angular velocity of an input shaft measured by an angle coder and a drag torque of an output shaft provided by a magnetic powder brake; B. calculating a collision time interval Δt_(i) between two gear teeth; C. comparing the collision time interval Δt_(i) with a time-step Δt, entering step D if Δt is less than Δt_(i); and entering step G if Δt is greater than or equal to Δt_(i); D. calculating a relative displacement x(n+1) by taking the time-step Δt as a step, if an absolute value |x(n+1)| of the relative displacement x(n+1) is less than or equal to a gear backlash L, entering step J; and if the absolute value |x(n+1)| of the relative displacement x(n+1) is greater than the gear backlash L, shortening the time-step Δt and then entering step E; E. comparing the time-step Δt with a minimum time-step t_(min), entering step D if the time-step Δt is greater than or equal to the minimum time length t_(min); and entering step F if the time-step Δt is less than the minimum time-step t_(min); F. calculating a relative displacement x(n+1)=Lsign({dot over (x)}(t(n))) by taking the minimum time-step t_(min) as a step, {dot over (x)}(n+1)=0 and then entering step J; G. calculating the relative displacement x(n+1) by taking the collision time interval Δt_(i) as the step, if an absolute value |x(n+1)| of the relative displacement x(n+1) is less than or equal to L−ε, entering step J; if the absolute value |x(n+1)| of the relative displacement x(n+1) is less than or equal to L+ε and the absolute value |x(n+1)| of the relative displacement x(n+1) is greater than L−ε, entering step I; and if the absolute value |x(n+1)| of the relative displacement is greater than L+ε, decreasing the collision time interval Δt_(i), and then entering step H; H. comparing the collision time interval Δt_(i) with the minimum time-step t_(min), if the collision time interval Δt_(i) is greater than or equal to the minimum time-step, returning to step G; if the collision time interval Δt_(i) is less than the minimum time-step t_(min), taking the minimum time-step as the step, i.e., x(n+1)=t(n)+t_(min), and then entering step I; I. calculating the relative displacement x(n+1)=Lsign ({dot over (x)}(t(n))), increasing the number of the step n by 1, then taking a collision time constant ∈_(Δt) as a step, i.e., t(n+1)=t(n)+∈_(Δt), calculating a relative velocity as: {dot over (x)}(n+1)=−e{dot over (x)}(n), calculating the relative displacement as: x(n+1)=x(n), and then entering step J; and J. judging whether t(n+1) is greater than M×T_(f), if yes, finishing calculation, and if not, increasing the number of step n by 1 and then returning to step B; ε being a minimum error, M being a number of the period, T_(f) being a period of the excitation, t(n) being a t(n)^((th)) time sampling point, and x(n) being an n^((th)) relative displacement.
 2. The method for determining dynamic transmission error of gear according to claim 1, wherein a calculation mode of the collision time interval in step B is as follows: ${{\Delta \; t_{i}} = \frac{{- b} \pm \sqrt{b^{2} - {4\; {ac}}}}{2\; a}},$ wherein, a=(R_(bg)T_(m)+R_(bg)T_(p) cos(ωt₀)−R_(bg)kx(T₀)−R_(bg)c{dot over (x)}(t₀))/2, b={dot over (x)}(t₀), c=x(t₀−Lsign({dot over (x)}(t₀)), Δt_(i) is a minimum positive real value of two values, R_(bg) is a base radius of a driven wheel, R_(bp) is a base radius of a driving wheel, T_(m) is a ratio of a mean value of an external load moment and a rotational inertia of the driven wheel, T_(p) is a ratio of a fluctuation portion of the external load moment and the rotational inertia of the driven wheel, ω is a frequency of the external load moment, t₀ is an initial system time, k is a ratio of meshing stiffness of contacted front and side of a gear pair caused by an oil film and the rotational inertia of the driven wheel, and c is a ratio of meshing damping of the contacted front and side of the gear pair caused by the oil film and the rotational inertia of the driven wheel.
 3. The method for determining dynamic transmission error of gear according to claim 1, wherein Δt_(i)<Δt is directly judged if Δt_(i) has no positive real value while calculating the collision time interval Δt_(i).
 4. The method for determining dynamic transmission error of gear according to claim 1, wherein a specific operation of shortening the time-step in step D is decreasing Δt² from Δt.
 5. The method for determining dynamic transmission error of gear according to claim 1, wherein a specific operation of shortening the collision time interval in step D is decreasing Δt_(i) ² from Δt_(i). 